#505678

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,16,21)(11,22,19,14)(12,25,29,36)(13,20,32,24)(15,33,31,18)(17,35,37,27)(23,34,26,40)(28,38,30,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,10,16,19,22,21)(12,26,28,37,29,23,30,17)(13,20,31,33,32,24,15,18)(25,27,39,40,36,35,38,34);(3,7)(5,8)(9,12)(10,17)(11,23)(13,27)(14,28)(15,34)(16,29)(18,36)(19,26)(20,38)(21,37)(22,30)(24,39)(25,33)(31,40)(32,35);(9,13,14,31,16,32,22,15)(10,18,19,20,21,33,11,24)(12,27,28,40,29,35,30,34)(17,36,26,38,37,25,23,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412874);